459
Notre Dame Journal of Formal Logic
Volume 30, Number 3, Summer 1989
Truth Table Logic,
with a Survey of Embeddability Results
NEIL TENNANT*
Abstract What logic is barely justified on the basis of the 'meanings' given
to the connectives by the left-right readings of their truth tables?
The valid arguments involved in truth table computations are called
Kalmaric. We set out a system T, consisting of normal proofs constructed
by means of elegantly symmetrical introduction and elimination rules. In the
system T there are two requirements, called (D) and (>), on applications of
discharge rules. T is sound and complete for Kalmaric arguments. (D) requires nonvacuous discharge of assumptions; (>) requires that the assumption discharged be the sole one available of highest degree.
We then consider a 'Duhemian' extension T*, obtained simply by dropping the requirement (>). T* is a proper subsystem of intuitionistic relevant
logic. Our main result is that T* is a double negation consistency companion to classical logic. Thus all one needs to add to T* to obtain classical logic
is the (intuitionistic) absurdity rule, and the (classical) rule of double negation elimination. T* represents the inferential core that is justified by the
left-right readings of the truth tables.
We survey all the embeddability results using various translation mappings "downwards" into subsystems of classical, intuitionistic, minimal, and
intuitionistic relevant logic. This puts our main result into significant context.
/ How does one read off a logic from truth tables?
It is often claimed that
the standard two-valued truth tables for —, &, v, and D capture the meanings
*Material from this paper was presented to the Conference of the New Zealand Division of the Australasian Association of Philosophy held in Auckland in August 1988;
to the Philosophy Seminars at the Universities of Wollongong and Adelaide; and to the
Philosophy Seminar in the Research School of Social Sciences, Australian National University. I am especially grateful to John Slaney and Timothy Smiley for their comments
on an earlier version.
Received February 13, 1989
460
NEIL TENNANT
of those connectives, and that those meanings are the classical ones. That is to
say, the logic justified by the meanings in question is classical logic.
This claim bears interesting closer scrutiny. What do the truth tables for
the connectives actually sayΊ They say that //the truth values of the components
of a compound sentence are such-and-such respectively, then the truth value of
the compound itself is so-and-so. I shall call these the left-right readings of the
truth tables; and I shall be interested in them, and them alone. There is a presupposition in these readings that truth values will be assigned uniquely, if at all.
That is, truth value assignments are functions, or many-one mappings. But the
truth tables do not also say that any sentence must be assigned either the value
T (True) or the value F (False) by any given assignment. That is, truth value
assignments need not be total functions (or, as I shall say, they need not be classical). If they are classical, this can only be shown by the statement of the truth
tables, insofar as T and F are the only values appearing within them. But this
aspect of what is shown is nowhere said explicitly in the left-right readings. If
we confine ourselves to what the truth tables say, it is possible, prima facie, that
some logic other than classical logic will emerge as the logic justified on the basis
of the meanings conferred on the connectives by the truth tables.
This paper realizes that possibility. It investigates two systems, which I shall
call T and T*, of truth table logic: a logic justified on the basis of what the truth
tables say, rather than on what they might arguably also show. T is the smallest
such system; T* is a very natural but modest extension of T. Their exact definitions will be given in due course.
2 Restricted and unrestricted transitivity
Both T and T* are subsystems of
the system of intuitionistic relevant logic (IR) developed in Tennant [17]. The
deducibility relations of T and T*, like that of intuitionistic relevant logic, fail
to be unrestrictedly transitive. That is, the following principle of unrestricted
transitivity does not hold:
If Pi is deducible from Xλ,
Pn is deducible from Xn,
and Q is deducible from Xo U
[Pί9...9Pn}9
then Q is deducible from XO\JXX U . . . U Xn.
But in the systems T* and IR the failures of this unrestricted transitivity principle are virtuous. Transitivity fails where, according to a relevantist, it ought
to fail: when the deducibility would otherwise hold only by virtue of the fact that
Xo U Xι U . . . U Xn is inconsistent, or Q is (intuitionistically) logically true. The
deducibility relations of the systems T* and IR satisfy instead the following,
epistemically superior, restricted transitivity principle:
If Pγ is deducible from X\,
Pn is deducible from Xn,
and Q is deducible from Xo U [Pu . . . ,Pn], then either Q or Λ (absurdity)
is deducible from some subset of Xo U Xx U . . . U Xn.
TRUTH TABLE LOGIC
461
This major structural difference between, on the one hand, the systems T*
and IR and, on the other hand, the unrestrictedly transitive systems of classical, intuitionistic, and minimal logic, is the reason for the interest attaching to
results about the embeddability of any of the latter into any of the former. In
particular, any result about the embeddability of classical logic (the largest unrestrictedly transitive system) into T * (the smallest restrictedly transitive system
under consideration) would be of particular interest. To appreciate this, note the
relative containments in Figure 1.
There are many well-known results about embeddability "downwards" in
the chain C, I, M of unrestrictedly transitive logics (or theories closed under
those logics); we will review these in a separate section. [17] gives a generalized
Godel-Glivenko theorem about the embeddability of C and of I into IR. In this
paper the main result concerns the similar embeddability of C into T *:
V
Main Theorem
or
7
Every classical proof P can be converted into a proof R
A
—A
~~Z
R (for some subset Z of X)
wf.
Λ
Classical
Intuitionistic
—A :A
A, -A : B
Minimal A,-A : ~B
:A DA
A
A
&B:A
&B:B
AD
(ADB).ADB
1
A,ADB:B
AM By ~A\B
~AV
B.ADB
Intuitionistic Relevant
Figure 1.
1
~A:ADB
Truth Table
462
NEIL TENNANT
This result tells us that the simplest translation in the book —prefixing with
double negations—maps C into T* "where it counts". Let us call a classically valid
argument perfect just in case its premises are satisfiable and no proper subset of
them logically implies the conclusion. A corollary to our main result is this:
(i) double negation maps perfect arguments into T*
(ii) if a set of sentences is not satisfiable, Γ* will provide a reductio from (some
subset of the set consisting of) their double negations, and
(iii) if a sentence is logically true, Γ* will provide a proof of its double negation.
In order to appreciate the significance of how T* thus contains the restrictedly
transitive fragment of classical logic under double negation, one has first to see
just how "small" the system T* is.
3 Rules of inference designed to mimic the truth tables: The systems T and T*
Any truth table logic has to be able, at the very least, to represent by means of
deducibilities the left-right evaluative transitions corresponding to the rows of
the truth tables. We need, that is, a sentential rendering of the inputoutput relations represented by the rows of each truth table. To that end, let us
write in place of every occurrence of T in a truth table an occurrence of the sentence to which that T is assigned; and in place of every occurrence of F in a truth
table an occurrence of the negation of the sentence to which that F is assigned.
(There will never be any confusion over whether a given occurrence of T stands
for the truth value or for the system of truth table logic soon to be defined.) Let
us then read the components' entries in any row of a truth table as the premises
of a simple argument whose conclusion is the compound's entry in that row.
Each such argument mimics the action of the truth table as represented by that
row; it captures exactly the left-right reading of the row. We shall use the letter p as a variable over rows, i.e., truth value assignments.
Each row in the tables below represents a truth value assignment, whose
truth set (formed from the set of atoms in question) we shall define as the set
consisting of those atoms (if any) to which the value T is assigned, along with
the negations of those atoms (if any) to which the value F if assigned. For any
sentence A, the truth set of p formed from A's atoms will be called p [A]. Truth
sets are obviously consistent, in that no truth set contains any atom along with
its negation.
In the diagram below, the truth tables are given in the left column. The
middle column consists of the simple arguments set up, row by row, according
to the method described above. The right column contains only those arguments
that really need to be independently established after taking care of redundancies in the middle column. We are taking a 'fell swoop' view of what the truth
tables say. Thus we read the truth table for disjunction as saying:
if the truth value of A is T then that of (A v B) is T;
if the truth value of B is T then that of {A v B) is T;
if the truth values of A and B are both F, then the truth value of {A v B)
isF.
TRUTH TABLE LOGIC
463
Truth table inferences
A
Sentential version
~A
T F
FT
A :. — A
~A Λ -A
A
B
A &B
T
T
T
T
F
FT
F
F
F
F
F
A
B
AMB
T
T
T
T
F
T
A,~B:.AvB
T
~A,B:.AVB
FT
F
F
F
A
B
ADB
T
T
T
F
FT
F
F
T
F
T
T
A,B:.A&B
A,~B:.~(A&B)
~A9B:.~(A&B)
Nonredundant demonstranda
AΛ —A
A9B:.A&B
~B ;.~(A &B)
~A :.~{A&B)
~A,~B .'. -(A & B)
A,B:.AVB
~A,~B :.~(AMB)
A,B:.ADB
A,~B :. ~(ADB)
~A,B Z.ADB
~A,~B .ADB
A:. AMB
B .AMB
~A,~B Λ -{AM B)
A,~B.. ~(ADB)
B Λ A DB
~A .. A D B
Note two important casualties of our way of reading the truth tables. First,
— A Λ A (the law of double negation) fails. Nothing that the truth table says
can guarantee that the falsity of ~A can arise only from the truth of A. Secondly, A,A D B .'. B (modus ponens) fails. The usual (classical) justification of
modus ponens points to the second row of the truth table for D: one cannot
have both A and ADB true with B false. But from this it follows only that if
both A and ADB are true, then it is not the case that B is false. And this we
shall recognize by the deducibility A,A D B Λ ~~B.
Kalmar's Theorem (sentential version)
For any truth value assignment p and
any sentence A
(i) if A is true under p, then from ρ[A] one can deduce A
(ii) if A is false under p, then from ρ[A] one can deduce ~A.
Kalmar's Theorem (inferential version)
For any truth value assignment p and
any sentence A
(i) if A is true under p, then from p [A] one can deduce A
(ii) if A is false under p, then from A,p[A] one can deduce Λ.
Let X be any subset of p[>l]. Then arguments of the form X:A or
X,A : Λ will be called simple. A simple valid argument will be called Kalmaric.
The deducibilities in the rightmost column of the table above, which I shall
call the list L, are just the ones we usually establish in order to prove (by indue-
464
NEIL TENNANT
tion on the length of sentences) the sentential version of Kalmar's Theorem for
the logic in question. One can also, however, recast these deducibilities in a
slightly more convenient form, in order to prove the inferential version of
Kalmar's Theorem. The variant list LΛ is
A,—A Λ Λ
A9B :. A &B
~B,A & B Λ Λ
-A,A & B Λ Λ
A :. A v B
B
.AvB
~A9~B,A V 5 : . Λ
A,~B9A D B Λ Λ
B
:.ADB
-A Λ A D B.
It is well-known that intuitionistic logic satisfies Kalmar's Theorem (in both its
sentential and inferential versions) in that it serves up all the deducibilities in the
union of the lists L and IΛ Note that all of these are simple (and Kalmaric) in
the senses defined above:
LULΛ
A Λ
A
A,-A Λ Λ
A,B
.A&B
~B,A & B Λ Λ
~A,A & B Λ Λ
~A Λ -(A &B)
~B Λ -(A &B)
A
.AvB
B
:.AvB
~A,~B,A V 5 : . Λ
^ r ί : . ~(AvB)
A9~B,A D B Λ Λ
A,~B . ~(ADB)
B :.ADB
-A .\ A D B.
Thus, intuitionistic logic suffices to mimic the process of evaluating a compound sentence as true or false, given the truth values of the atoms occurring
within it. The extra strength of classical logic enters into the picture when we
go beyond Kalmar's Theorem to show that every sentence true in every row of
its truth table is a theorem (that is, deducible from the empty set of assumptions). By Kalmar's Theorem any such sentence can be deduced from each of
the possible truth sets. But in order to have a proof of that logically true sentence from the empty set of assumptions one must appeal to some classical rule
such as dilemma, which allows one systematically to discharge those atoms and
their negations that occur as assumptions within the proofs served up by
Kalmar's Theorem.
TRUTH TABLE LOGIC
465
What is less well-known, and indeed is rather obvious, is that a much
weaker sublogic of intuitionistic logic still suffices for Kalmar's Theorem. The
logic in question can simply be taken to be the closure (under cut) of the deducibility schemata in the list L U LΛ.
But this way of specifying the weakest "Kalmar logic" is not very satisfying. Trivial though the deducibility schemata in L U LΛ may be, they nevertheless form a rather ad hoc collection from the proof-theoretic point of view. What
would be preferable is a specification of introduction and elimination rules for
the logical operators, taken one by one and in isolation.
Question
How can Λve frame rules of natural deduction for the smallest logic
(call it T) for which both the sentential and inferential versions of Kalmar's Theorem hold?
(In what follows, I shall mean by "Kalmar's Theorem" both its sentential
and inferential versions.)
Note that in each deducibility schema in the list L U LΛ the degree of its
conclusion, if other than Λ, exceeds the highest degree of its premises. It follows
that any deducibility in the closure of these schemata has the same property.
Thus the deducibility A & B Λ A will not be contained in the logic for which
we are about to provide rules of natural deduction. Nor, as remarked earlier,
will ~~A Λ A or A,A D B Λ B.
With an eye to the list L U LΛ, and to the requirements of Kalmar's Theorem, we see that within the logic T we have to be able to prove
(i) at least some consequences of consistent sets
and
(ii) at least some inconsistencies.
A proof is said to be simple just in case its set of undischarged assumptions,
along with its conclusion, form a simple argument. A proof is in normal form
(or normal) just in case no sentence occurrence within it stands as the conclusion of an application of an introduction rule and as the major premise of an
application of the corresponding elimination rule. A normal proof makes no
unnecessary "detours" in leading one from its undischarged assumptions to its
conclusion.
In order to generate the deducibility schemata in L U LΛ, and the deducibilities required in general by Kalmar's Theorem, I now set up the system T of
truth table logic.
Definition of T
The system T of truth table logic consists of normal proofs
built up by means of the following rules. Introduction rules appear on the left;
the corresponding elimination rules appear on the right. Note that the absurdity
rule is absent. In the statement of these rules, D> represents a dual requirement:
(D)
There must be an undischarged assumption of the indicated form, available for discharge by the application of the rule in question
(>)
That assumption must be the sole undischarged assumption of highest
degree within the subproof in question.
466
NEIL TENNANT
Note further the pleasing symmetry between introduction and elimination
rules. Introduction rules tell one how to reason towards a complex conclusion;
elimination rules tell one how to reduce a complex assumption to absurdity.
Introduction rules
Elimination rules
•>4(i)
-——
Λ
Ά
TV*
&
Λ
*
D>
TVS
(/)
A
T(/)
D>
D >
A
ADB0)
r n
B
A^~B
ADB
T(/)
B
Λ
A
AvA_A_
D>—(/)
&
(/)
¥(/)
A (/)
D >
^(/)
Λ
A
Λ
(/)
Reminder: Applications of these rules in T are always subject to the overall
requirement that the resulting proof be normal. So one cannot apply an introduction rule and then immediately afterwards apply the corresponding elimination rule. Therefore, given that all elimination rules have Λ as conclusion, we
have:
Lemma 1
In any proof in the system of truth table logic, every major premise for an elimination stands alone, with no sentence occurrences above it.
By inspection of the rules, and induction on the complexity of proofs, we also
have:
Lemma 2
Every proof in the system T of truth table logic is simple; that is,
the argument it proves is simple.
This immediately yields what is in effect a Kalmaric soundness theorem:
every argument provable in the system T of truth table logic is Kalmaric, that
is, simple and valid. Kalmar's Theorem now in effect becomes a Kalmaric completeness theorem for the system T of truth table logic: every Kalmaric argument
can be proved in T.
TRUTH TABLE LOGIC
467
Let us now see how the proof of Kalmar's Theorem proceeds for the system T. Note that in the presence of the introduction and elimination rules for
- just given, it suffices to establish only the inferential version of Kalmar's
Theorem.
Kalmar's Theorem (inferential version)
For any (total classical) truth value
assignment p and any sentence A
(i) if A is true under p, then there is a Ί-proof of A from p[A]
(ii) if A is false under p, then there is a Ί-proofofΛ from A,p[A],
Proof: By induction on the length of sentences A.
Basis: If an atom A is true under p then p[A] = {A} whence, trivially, there
is a T-proof of A from p[A]. If on the other hand A is false under p then
p[A] = {-A}, whence a single step of —Elimination is a T-proof of Λ from
A9p[A].
Inductive Hypothesis: Assume that the result holds for all sentences less complex (that is, of lower degree) than A.
Inductive Step: Consider A by cases.
(~)
A is of the form ~ B:
Suppose that A is true under p. Then B is false. By the inductive hypothesis there is a T-proof of Λ from Byp [B]. By —-Introduction we extend this to
a T-proof of ~B from p[B]; i.e., of A from p[A].
Now suppose that A is false under p. Then B is true. By the inductive
hypothesis there is a T-proof of B from p [B]. Now assume that ~B and apply
—Elimination to obtain a T-proof of Λ from ~ B9p[B]; i.e., from A,ρ[A].
(&) A is of the form (B & C):
Suppose that A is true under p. Then B is true and C is true. By the inductive hypothesis there is a T-proof of B from p[B] and a T-proof of C from
p[C], Now apply &-Introduction to obtain a T-proof of B & C from p[B & C].
Now suppose that A is false under p. Then either B is false or C is false.
Without loss of generality suppose that B is false. Then by the inductive hypothesis there is a T-proof of Λ from B,p[B], Now apply &-Elimination to obtain
a T-proof of Λ from B & C,p [B]:
The result is a T-proof of Λ from B & C,p [B].
(v)
A is of the form (B v C):
Suppose that A is true under p. Then B is true or C is true. Suppose without loss of generality that B is true. Then by the inductive hypothesis there is
a T-proof of B from p [B]. Now apply v-Introduction to obtain a T-proof of
By C from p[B].
468
NEIL TENNANT
Now suppose that A is false under p. Then B is false and C is false. By the
inductive hypothesis there is a T-proof of Λ from B,p [B] and a T-proof of Λ
from C,p[C]. Now apply v-Eϋmination to obtain a T-prόof of Λ from B v
C9p[B],p[C] (i.e., from £ v C , p [ £ v C ] ) .
(D) A is of the form (B D C):
Suppose that A is true under p. Then B is false or C is true. Suppose first
that B is false. Then by the inductive hypothesis there is a T-proof of Λ from
B,p [B]. Now apply D-Introduction to obtain a T-proof of B D C from p [B].
Suppose next that C is true. Then by the inductive hypothesis there is a T-proof
of C from ρ[C]9 which can be extended by D-Introduction to obtain a T-proof
of BD C from p [ C ] .
Now suppose that A is false under p. Then B is true and C is false. Thus
by the inductive hypothesis there is a T-proof of B from p[B] and a T-proof
of Λ from C,p [C]. By D-Elimination there is a T-proof of Λ from B D C, p [B],
and p[C] (i.e., from B D C,p[£ D C]):
—
p[£]
T^
B^C
(I)
C,p[C]
A
Λ
This completes the proof of Kalmar's Theorem for the system T of truth table
logic.
We shall now investigate how one might relax the requirement of simplicity
in order to obtain more reasonable closure than we have in T. Recall the dual
requirement on any application of a discharge rule in the system T:
(•)
there must be an undischarged assumption of the indicated form, available for discharge by the application of the rule in question
(>)
that assumption must be the sole undischarged assumption of highest
degree within the subproof in question.
Note that although we have the T-proof
<»A
~Λ
^
A&B
Λ
the requirement (>) on discharge rules allows us to extend it to
(y 2 , —
'A
&B
0
>
Λ_
(2)
^
j
TRUTH TABLE LOGIC
469
but not to
AAΆ
^
But surely, insofar as truth-preservation is concerned, it does not matter which
of the premises in a reductio one chooses for subsequent discharge and denial?
Note that the requirement (>) concerns only subproofs having the form of a
reductio. Therefore this Duhemian objection to the overly restrictive nature of
(>) can be pressed quite generally.
The most obvious relaxation is therefore to drop the requirement (>) but
still to retain the requirement (D) and the requirement that proofs be normal.
We thereby arrive at what might be called the Duhemian extension of the system T, which I shall call T*.
Definition of T*
Let T* be the system, based on the same rules as given
above for T, that results from dropping the requirement (>) but still retaining
the requirement (D) and the requirement that proofs be normal.
An immediate difference between T and T* is that in T* one can now prove
. -(A & B). It should also be obvious that T* contains T.
An interesting pathology of T* (hence also of T) is that it lacks AD A as
a theorem. One sees this by inspection of the (two halves of the) D-Introduction
rule. Yet there is the following proof in T (hence in T*) of —
(ADA):
-A
_jθ)
ADA
-(ADA)*®
A~5A~{1)
~(ADA)i2)
Likewise, T* does not contain disjunctive syllogism: A v B9 ~A .'. B, nor the
usual forms of &-elimination: A & B Λ A and A & B .'. B. But T (and hence
T*) does contain Aw B, -A Λ — B , and, as we saw above, A & B Λ — A
and A & B Λ ~~B.
Although our introduction and elimination rules for D in the systems T and
T* have been framed so as to make AD B look as though it ought to be equivalent to ~ A v B, these two formulas nevertheless fail to imply each other in T*
(hence also in T). Note that in intuitionistic logic (and even in intuitionistic relevant logic) ~ A v B implies A D B, though not conversely. To see that —A v B
nevertheless does not imply A D B in T*, note that ~A v B does not imply B,
and is consistent with A. Thus neither half of the rule of D-Introduction in T*
can be invoked to produce A D B as the conclusion of a proof with —A v B as
its only undischarged assumption.
In T* the contraction inference AD (AD B): AD B also fails. Note that
470
NEIL TENNANT
in any proof Π in T (T*) the subtree subtended by any sentence occurrence
within Π is itself a proof in T (T*). That is to say, if one prunes away a subtree determined by a sentence occurrence within Π, what comes away is a proof.
But what is left of Π (even after restoring the sentence occurrence in question)
might not be. For example, in T* we have the following proof:
A ΊF
A&B
(1)
~(A&B)
The subtree determined by the occurrence of A & B is a T*-proof:
A B
A&B.
But what is left of the original T*-proof after the latter has been pruned away,
even after restoring the occurrence of A & B, is not a proof in T*:
A&B
-(A&B)
Λ
n
because there is no occurrence within it of B as an assumption to be discharged
by the final step of —Introduction.
This highlights the difference between two kinds of closure:
(I) Any pruned-away fragment of a proof is a proof.
(II) The residue of any pruning-away of a fragment from a proof is a
proof.
Systems such as T and T*, and the relevant systems of classical and intuitionistic
logic developed in [14] and [17], satisfy only (I). By contrast, the systems of classical, intuitionistic, and minimal logic satisfy both (I) and (II).
4 Proof theory for T and T*
T* contains T. The main feature of proofs in
T* is that they have to be in normal form, and must actually contain assumption occurrences as indicated by the various discharge rules. In this regard we
shall call T* a tight system. T, obviously, is also tight. T, T*, and the relevant
systems of intuitionistic and classical logic just mentioned are moreover what
I shall call trim, in that they lack the absurdity rule (exfalso quodlibet):
Λ
A.
These observations motivate the following definitions of various changes one
can make to a given proof system framed in terms of introduction and elimination rules:
TRUTH TABLE LOGIC
471
One tightens by requiring proofs to be in normal form and making assumption
occurrences obligatory; one trims by banning the absurdity rule.
Conversely, respectively:
One slackens by dropping the requirements of normality and obligatory assumption occurrence; one bloats by adding the absurdity rule.
I want eventually to prove the result promised above about T*, that T* is
a double negation consistency companion to classical logic in the sense of the
Main Theorem
or
X
—Z
Every classical proof P can be converted into a proof R
A
—A
—Z
R (for some subset Z of X) in T*.
Λ
In this respect T* is like the system IR of intuitionistic relevant logic presented
in [17]. But it is much weaker than IR, while yet being a double negation consistency companion to classical logic. It also has D as a primitive connective, like
the modified system of IR investigated in [18].
T* is thus a proper subsystem of IR. (It contains —A .'. ADB, however,
so it is not a subsystem of minimal logic.) Despite the fact that T* is a proper
subsystem of IR, however, if we simply add the rule of double negation elimination to T* we produce a system of classical relevant logic that matches classical logic on all consistent sets of premises, and proves all inconsistencies. This
is a striking corollary of our main theorem, which we shall prove shortly.
Even though intuitionistic logic is a double negation companion to classical
logic, one cannot hope to achieve our main result—that the system T* of truth
table logic is a double negation consistency companion to classical logic—by simply establishing that T* is a consistency companion to intuitionistic logic (thereby
dividing the labor between two transitions, the first from C to I, the second from
I to T*). For T* is not a consistency companion to intuitionistic logic, as the
nontheoremhood in T* of A D A dramatically shows.
We proceed now to the proof of the main theorem. First we recall two
results, but framed in the new terminology just defined, due to Prawitz [11] and
myself [14] respectively:
Normalization Theorem
Every proof of A from X in a given slack and
bloated system can be converted into a proof in normal form of A from (some
subset of) X in that slack and bloated system.
Extraction Theorem
Every proof in normal form of A from X in the bloated
slackening of a given trim and tight system S can be converted into a proof of
A or of A from (some subset of) X in the system S.
In order to build on these to obtain our main theorem we need two more
theorems. Theorem 1 below draws on the normalization and extraction theorems. Theorem 2 is established independently.
472
Theorem 1
NEIL TENNANT
X
Every proof P in slackened bloated T* can be converted into
A
Z
Z
a proof R or R (for some subset Z of X) in T*.
A
Λ
Proof: First we normalize P by means of the reduction procedures to obtain a
Y
proof Q (for some subset YofX) in normal form within slackened bloated T*.
A
Z
Z
Then we extract a proof R or R (for some subset Z of Y) in T*.
A
A
Theorem 2
In the language based on ~, &, v, and D every classical proof
X
—X 1
—X f
P can be converted into a proof P' or P' (for some subset X' of X) in
A
— A
Λ
slackened T*.
Proof: By induction on the length of classical proofs. The method is exactly that
of the proof of the generalized Godel-Glivenko theorem for intuitionistic relevant logic given in Chapter 24 of [17]. One only has to take a little care to establish that the required transforms are indeed available in slackened T*.
Basis: Obvious.
Inductive Hypothesis: Assume that the result holds for all proofs less complex
than P.
Inductive Step: Consider P by cases, according to the rule applied in the last step
of P. Remember we are now dealing with proofs P constructed in accordance
with the standard rules of inference (as in [11] or [13]) for classical logic. In particular, we do not require discharge rules actually to discharge assumptions of
the indicated form. Moreover, the more familiar introduction and elimination
rules for D (conditional proof and modus ponens) are in use, as well as both the
absurdity rule and the classical rule of reductio.
In each case below, the form of P is given in bold. Below it are given the
forms of the transform P' in slackened T* as desired, depending on the form
that might be taken by the transforms IT in slackened T* that are guaranteed
by the inductive hypothesis for the immediate subproofs Π of P.
Case 1: P ends with an application of classical reductio:
π
TRUTH TABLE LOGIC
473
Then P' will have one of the following two forms, depending on the form of IT:
Π'
or
Y'
IT
Λ
.
Case 2: P ends with an application of the absurdity rule:
X
Π
Λ
Then P' will have the form:
w
Λ
.
Case 3: P ends with an application of —Introduction:
π
^>.
Then P ' will have one of the following two forms, depending on the form
of Π':
Λ(1)
Y'
Π'
or
Π'
Λ
.
474
NEIL TENNANT
Case 4: P ends with an application of —Elimination:
Y
Π
A
Z
Σ
~A
Λ
Then P' will have one of the following three forms, depending on the forms of
IT andΣ':
Π'
Σ'
.,_, Λ
Λ
Λ
or
Π'
Λ
or
—Z'
Σ'
Λ
Case 5: P ends with an application of &-Introduction:
Y
Z
Π
Σ
A
B
A &.B.
Then P' will have one of the following three forms, depending on the forms of
Π' and Σ':
(1)
(2)
:r τ
—
A&B
(3)
~{A&B)κt
—Y
W
—Λ^
Z-
Λ
~By)
~j~B
—
or
Y
Π'
Λ
(A&B)^
TRUTH TABLE LOGIC
475
or
Σ'
Λ
Case 6: P ends with an application of &-Elimination:
X
Π
A&B
A .
Then P' will have one of the following two forms, depending on the form of Π':
A&BK
Λ
'
—X'
Π
t
(j)
w
~(A &B)
— ( A &B)
or
IT
Λ
Case 7: P ends with an application of v-Introduction:
X
Π
A
AM
B.
Then P' will have one of the following two forms, depending on the form of IT:
τ(1)
AMB
~(AVB)
V ;
—X'
Π
— (i)
~AK)
'
—A
~~MvH)g)
or
X'
Λ
476
NEIL TENNANT
Case 8: P ends with an application of v-Elimination:
Z,T(/)
Σ
Y
Π
W,~B(Π
H
<?(/)
ά^M—S.
Then P' will have one of the following nine forms, depending on the forms of
Π', Σ ' . a n d S ' :
Σ
(5) —
K
AvB
Λ
'
S
(5)—
'
'
Λ
Λ
— F
Π
( 4 )
or
i 4 v J
W
Λ
Λ_
A
-(A
Y'
( 3 >
Π'
( 4 )
MB)
K }
— ( A v B)
or
(3)
T ^4
~~Z'9
(5)
(4)
ττ/f
zz£
(4)
~ U Ί By
Λ
A
yf
s±
( 1 )
}
'
^ ^ 0
,y\
^ '
*
Π
—(A vB)
TRUTH TABLE LOGIC
477
or
Λv£
Σ'
Λ
W
S'
Λ_
Y'
^ '
A
TT'
(4)
Λ
or
—γ>
IT
A
or
or
Σ'
—C
Σ'
Λ
or
or
—W>
E'
—w'
S'
C
Λ
.
Case 9: P ends with an application of D-Introduction:
π
Then P ' will have one of the following four forms, depending on the form of
IT:
¥(2)
A DB
^r^
~(ADB)
Λ
(2)
^B
W
Y',
A
Π'
~~B
(
~~MD^)(4)
'
478
NEIL TENNANT
or
—^—(1)
IT
or
τ
(1)
ΰ
(2)
K
APB
~(ADB)
'
~BK '
Y'
—B
~~04 Λ W 2 )
or
~*γ'
Π'
Λ
.
Case 10: P ends with an application of D-Elimination:
Y
Π
A
Z
Σ
APB
B
Then P' will have one of the following three forms, depending on the forms of
r
IT and Σ :
5
;
-ir
Λ
^4(1)
^
—Y'
Π'
— ^
~~Z'
(2)
Σ>
^ZBW
or
Π'
Λ
TRUTH TABLE LOGIC
479
or
Σ'
Λ
.
This completes the proof of Theorem 2.
Finally, we have our:
Main Theorem
X
—Z
Every classical proof P can be converted into a proof R
A
or
R
~~
Λ
(for some subset Z ofX) in T*.
Λ
Proof: Apply the method of Theorem 2 and then that of Theorem 1.
In the light of our main result, we see that T* satisfies two rather nice new closure conditions, generalizing the conditions, noted earlier, that in T we had to be
able to prove:
(i) at least some consequences of consistent sets
(ii) at least some inconsistencies.
The two new closure conditions are, respectively, that the logic deliver:
(i') transitivity of deducibility under consistent accumulation of premises
(which I shall call the consistent transitive closure condition) and
(ii') deducibility of all inconsistencies
(which I shall call the inconsistency closure condition).
The satisfaction of each of these conditions follows immediately from the main
theorem.
5 Comparison with other embeddability results
The literature contains many
examples of embeddings of one system into another by means of schematically
definable translations. In this section we survey all of these, and summarize the
results so far obtained. This gives the context that imparts the special interest
to our main result above.
Abbreviations of systems
C
I
M
MR
IR
T*
CA (IA)
CZF
IZF\E
is
is
is
is
is
is
is
is
is
classical logic
intuitionistic logic
minimal logic
minimal relevant logic
intuitionistic relevant logic
the system of truth table logic of this paper
classical (intuitionistic) arithmetic
classical Zermelo-Fraenkel set theory
intuitionistic Zermelo-Fraenkel set theory minus the axiom of extensionality.
480
NEIL TENNANT
^
1
s is
ϊ
I
! ί
m
^
.
j-
i
ί
>
>
°
I
I
I
°S
§ 5
I I
! !
δ
)
|
S
>
)
W
I
ί
£
ί ^
T
-έ
^
is x
it
I ^ I
I J ? f I 3? ^ I ί
ϊ
S
s°
1
i
S ί ^ ^
T T if f
2
^
,
J
£
S S
iS
"8
^
.2
+
^ ^
»
I
i i H
5
5
Ί "B ^
'
' '
CO
<
1^
£*
is
« n
^
3
I
»
n
& ui
m
TRUTH TABLE LOGIC
481
If/is a mapping from the language L into itself, we say that /distributes
over @ iff
f(A @B) = (f(A)) @ (f(B)) and f(QxΛ) = Qxf(A).
We list in tabular form on p. 480 the various translation mappings that have
appeared in the literature. Whenever the entry # appears the translation mapping is understood to be distributive for the case in question. The sentence S to
be translated can have one of the forms shown, with respect to which the translation mappings can be specified as given.
Forms of claims
f: U, V
means "/maps deducibility in U to deducibility in F "
f: (U, V) means "/maps theoremhood in Uto theoremhood in F "
f: [ I/, V] means "if A is deducible in U from X, then either fA or ± is
deducible in Ffrom some subset offX".
(In the terminology used earlier, we could also say that V is an f consistency
companion to U. Note that in terms of sequents, we could also say "if the
sequent X: Y is provable in £/, then for some subsequent X': Y', fXf :fY' is
provable in F " ; or, thinking of concentration as the converse of dilution, we
could say that / maps every deducibility in U to some (possibly more concentrated) deducibility in F; or, /concentrates deducibility in U to deducibility in
V.)
/ : U, V - { . . . }
means "in the language without the operators . . .,/maps
deducibility in U to deducibility in V"
f [U, V] — { . . . } means "in the language without the operators . . ., /concentrates deducibility in U to deducibility in F".
Note that when the target system Fis relevant, in the sense that it does not contain the first Lewis paradox A, -A .'. B, and the source system is not, the best
results we can hope for in general are of the form/: [ U, V] rather than/: U9 V.
Results
Sources
K:C,I
Kolmogorov 1925 [8] (propositional
logic only)
Glivenko 1929 [5] (propositional
logic only)
Gόdel 1932-3 [6] (propositional
logic only)
Gδdel 1932-3 [6]
Gόdel 1932-3 [6]
Gentzen 1934 [3]
Gentzen 1936 [4]
Kleene 1952 [7]
Kleene 1952 [7]
= :(C,I)
id: (C,I) - (v,D)
'(C,I)
':CA,IA
G: C,I
o: CA,IA
o Λ :CA,IA
r
* :CA,IA
482
NEIL TENNANT
if S is free of v,3 then S
is deducible in IA from — S
= : C , I - {V}
= : CA,IA - (V)
id: CA,IA - {v,3}
if all atomic constituents
other than those that are
antecedents of implications
are negated, then id: C,I
c:(C,M)
if S has no negative occurrence of
v and ~ S is a classical
theorem then ~ S is an
intuitionistic theorem
K:C,M
G:C,I
x
: (I,M)
F : (C,I)
Fα : CZF, IZF\E
= :C,M - {V,D}
t:C,I
t : C,M - [D]
P:I,M
D : I,M
id: [I,IR] - {D}
= : [C,MR] - {v,D,V}
= : [C,IR] - {D,V}
t : [C,MR] - {v,D}
t : [C,IR] - {D}
id:[I,IR]
= : [C,IR] - {V}
t:[C,IR]
Kleene
Kleene
Kleene
Kleene
1952
1952
1952
1952
[7]
[7]
[7]
[7]
Kleene 1952 [7]
Church 1956 [1]
Mine and Orevkov 1963 [10]
Prawitz-Malmnas 1968 [12]
Prawitz-Malmnas 1968 [12]
Prawitz-Malmnas 1968 [12]
Friedman 1973 [2]; via Kleene 1952 [7]
Friedman 1973 [2]
Tennant 1978 [13]
Tennant 1978 [13]
Tennant 1978 [13]
Leivant 1985 [9]
Leivant 1985 [9]
Tennant 1987 [17]
Tennant 1987 [17]
Tennant 1987 [17]
Tennant 1987 [17]
Tennant 1987 [17]
Tennant 1988 [18]
Tennant 1988 [18]
Tennant 1988 [18]
The main result (for propositional logic) of this paper is:
= : [C,T*].
It says that the simplest translation concentrates the largest system into the
smallest (relevant) one.
6 The question of a semantics
Our inquiry had a "semantical" starting
point —the left-right readings of the truth tables —but ended with purely
proof-theoretical results. We make no apology in reply to any objection that a
semantics is somehow "missing" for the system T*. For the whole burden of the
analysis has been to discover just what the supposedly semantic nature of the
logical constants, as "given" by the truth tables, really consists in. The general
TRUTH TABLE LOGIC
483
philosophical conviction behind the enterprise is that the intuitions of semanticists really only arise from the manipulations they would be disposed to make
in syntax, that is, from the rules of inference and of proof according to which
they would make their deductive transitions. (For a more extended defense of
this standpoint, see [16].) Our inquiry has focused on the question of what
exactly these transitions ought to be taken to be, if one is simply given the truth
tables as a supposedly "semantical" account of the workings of the logical connectives.
If the reader were to insist on an orthodox "semantics", in terms of which
soundness and completeness results could be given, then let it be the one given
earlier. Focus on the simple sequents, namely those of the form p[A] :A or
ρ[A] ,A : -L. Now consider those among them that are valid, that is, the Kalmaric sequents. These are the valid arguments used in calculating the truth value
of A under the various assignments p. The proof system T given above captures
exactly the Kalmaric sequents.
Dropping the discharge requirement (>) on proofs in the system T, thereby
getting the system T*, corresponds to widening one's interest in a Duhemian
spirit so as to include all premises of reductio proofs as candidates for discharge.
This widening takes place without changing the fundamental "semantical" character of the logical connectives, as given by the left-right readings of their truth
tables, and as now captured by the rules of proof for the system T*. The thought
was that if T* were the right syntactic story to be told about the connectives thus
given, we should be able to show that we had got hold of the uncorrupted core
of the slack and bloated "understanding" of the connectives claimed by the classicist. We have seen that double negation concentrates classical deducibility in
the system T*, thereby showing that the classicist is only two illicit steps beyond
the pale. First, he or she uses exfalso quodlibet, which, in the terminology developed here, shows a gross lapse of concentration; and secondly, he or she uses
double negation elimination (or some equivalent): a metaphysical article of faith
which is all Wittgensteinian show but which enjoys no truth-tabular go.
7 A further application of truth table logic
Quantified versions of T and T*
can be defined in the obvious way. The adoption of potentially infinitary versions of V-Introduction and 3-Elimination for T yields a system that enables one,
in a perfectly precise way, to account for the way in which basic facts about the
world can render a complex sentence true, or render it false. This enables one
to reformulate Ayer's celebrated and ill-fated criterion of verifiability in a most
satisfactory way. The explication is consonant with the logical empiricist's original motivating intuitions, and is immune to the Church-type collapses that have
bedeviled various attempts to refine Ayer's criterion. This application of the system T, however, is material for a future paper.
REFERENCES
[1] Church, A., Introduction to Mathematical Logic, Vol. 1, Princeton University
Press, Princeton, New Jersey, 1956.
484
NEIL TENNANT
[2] Friedman, H., "The consistency of classical set theory relative to a set theory with
intuitionistic logic," The Journal of Symbolic Logic, vol. 38 (1973), pp. 315-319.
[3] Gentzen, G., "Untersuchungen ϋber das logische Schliessen," Mathematische Zeitschrift, vol. 39 (1934), pp. 176-210.
[4] Gentzen, G., "Die Widerspruchsfreiheit der reinen Zahlentheorie," Mathematische
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[5] Glivenko, V., "Sur quelques points de la logique de M. Brouwer," Academie
Royale de Belgique, Bulletins de la Classe des Sciences, series 5, vol. 15 (1929), pp.
183-188.
[6] Gόdel, K., "Zur intuitionistischen Arithmetik und Zahlentheorie," Ergebnisse eines
mathematischen Kolloquiums, vol. 4 (1932-33), pp. 34-38.
[7] Kleene, S. C , Introduction to Metamathematics, van Nostrand, Princeton, New
Jersey, 1952.
[8] Kolmogorov, A., "Sur le principe de tertium non datur," Recueil Mathematique
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[9] Leivant, D., "Syntactic translations and provably recursive functions," The Journal
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[10] Mine, G. E. and E. P. Orevkov, "An extension of the theorems of Glivenko and
Kreisel to a certain class of formulas of predicate logic," Doklady Akademiia Nauk
SSSR, vol. 152 (1963), pp. 553-554.
[11] Prawitz, D., Natural Deduction: A Proof-Theoretical Study, Almqvist and Wiksell, Stockholm, 1965.
[12] Prawitz, D. and P. E. Malmnas, "A survey of some connections between classical, intuitionistic and minimal logic," pp. 215-229 in Contributions to Mathematical Logic, edited by A. Schmidt, K. Schΐitte, and H. J. Thiele, North Holland,
Amsterdam, 1968.
[13] Tennant, N., Natural Logic, Edinburgh University Press, Edinburgh, 1978.
[14] Tennant, N., "A proof-theoretic approach to entailment," Journal of Philosophical Logic, vol. 9 (1980), pp. 185-209.
[15] Tennant, N., "Perfect validity, entailment and paraconsistency," Studia Logica,
vol. 43 (1984), pp. 179-198.
[16] Tennant, N., "The withering away of formal semantics,"MindandLanguage, vol. 1
(1986), pp. 302-318.
[17] Tennant, N., Anti-Realism and Logic, Clarendon Press, Oxford, 1987.
[18] Tennant, N., "Intuitionistic mathematics does not need ex falso quodlibet," unpublished.
Department of Philosophy
Faculty of Arts
Australian National University
Canberra, ACT2601
Australia